Operations

Addition: of two vectors a + b is geometrically achieved by connecting the tail of the vector b with the head of a, Fig. 1.2. Analytically the sum vector will have components [ a\ + b\ a2 + b2 a3 + b3 J.

Figure 1.2: Vector Addition u+v

Figure 1.2: Vector Addition

Scalar multiplication: aa will scale the vector into a new one with components [ aai aa2 aa3 J. Vector Multiplications of a and b comes in three varieties:

Dot Product (or scalar product) is a scalar quantity which relates not only to the lengths of the vector, but also to the angle between them.

where cos 9 (a, b) is the cosine of the angle between the vectors a and b. The dot product measures the relative orientation between two vectors. The dot product is both commutative ab=ba

The dot product of a with a unit vector n gives the projection of a in the direction of n. The dot product of base vectors gives rise to the definition of the Kronecker delta defined where

Cross Product (or vector product) c of two vectors a and b is defined as the vector as c = axb = (a263 - a362)ei + (a3&i - ai63)e2 + (ai62 - a^bi)e3

which can be remembered from the determinant expansion of

ei

e2

e3

ax b =

ai

a2

a3

bi

62

and is equal to the area of the parallelogram described by a and b, Fig. 1.3.

Parallelogram
Figure 1.3: Cross Product of Two Vectors

The cross product is not commutative, but satisfies the condition of skew symmetry axb = -bxa (1.13)

The cross product is distributive aax (flb + 7c) = a^(axb) +a7(axc) (1.14)

Triple Scalar Product: of three vectors a, b, and c is desgnated by (axb)-c and it corresponds to the (scalar) volume defined by the three vectors, Fig. 1.4.

Figure 1.4: Cross Product of Two Vectors

ax

ay

az

bx

by

bz

(1.16)

cx

cy

Cz

The triple scalar product of base vectors represents a fundamental operation

1 if (i,j, k) are in cyclic order 0 if any of (i,j, k) are equal — 1 if (i,j, k) are in acyclic order

The scalars £ijk is the permutation tensor. A cyclic permutation of 1,2,3 is 1 ^ 2 ^ 3 ^ 1, an acyclic one would be1 ^ 3 ^ 2 ^ 1. Using this notation, we can rewrite c = axb ^ ci = £ijkajbk

Vector Triple Product is a cross product of two vectors, one of which is itself a cross product.

and the product vector d lies in the plane of b and c.

0 0

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